{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-finite-integral-domain"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-integral-domain-01","title":{"zh-Hant":"有限整環：以完整乘法表判定 ℤ/5ℤ 與 ℤ/6ℤ，再由有限性推出可逆","en":"Finite integral domain: deciding ℤ/5ℤ and ℤ/6ℤ from complete tables, then deriving invertibility from finiteness"},"summary":{"zh-Hant":"以完整乘法表判定 ℤ/5ℤ 是有限整環、ℤ/6ℤ 不是；再把「每個非零元可逆」作為定義之後的定理，對任何有限整環用有限性證明，並在 ℤ/5ℤ 中求出全部逆元，同時說明 ℤ/6ℤ 中 2 為何沒有逆元。","en":"Using complete multiplication tables, decide that ℤ/5ℤ is a finite integral domain and ℤ/6ℤ is not; then prove, as a theorem that comes after the definition, that every nonzero element of any finite integral domain is invertible, using finiteness; find all inverses in ℤ/5ℤ and explain why 2 has no inverse in ℤ/6ℤ."},"mko_ids":["mko-finite-integral-domain"]}],"title":"概念推導導讀"}